Physical Foundations of Cosmology

2.3 Conformal diagrams 53

I+

π/^2

I

i^0

χ

r= const

t= const

t= 0

i+

i−

π/^2

−π/ 2

I

−

η

Fig. 2.7.

Open and flat universesNow we will use the Minkowski conformal diagram to
construct the diagram for open and flat universes dominated by matter satisfying
the strong energy dominance condition,ε+ 3 p>0. The metric is

ds^2 =a^2 (η ̃)

(

dη ̃^2 −dχ ̃^2 − 2 (χ ̃)d 

2

)

, (2.43)

where the scale factoravanishes at a singularity occurring at ̃η= 0 .(Here we have
added tildes to the notation in (2.2) sinceη,χare reserved for coordinates with
finite ranges.) The conformal time ̃ηis confined to the range( 0 ,+∞).Since (χ ̃)
is equal to ̃χfor a flat universe and to sinh ̃χfor an open universe, in both cases
χ ̃changes from 0 to+∞.For ̃η> 0 ,the temporal–radial part of metric (2.43) is
related to the Minkowski metric (2.40) by a nonsingular conformal transformation.
The coordinatestandrconsidered in the upper half of Minkowski spacetime(t> 0 )
span the same range as the ̃η,χ ̃coordinates. Hence, the conformal diagrams of open
and flat universes should have the same shape as the upper half of the Minkowski
conformal diagram (Figure 2.8). The hypersurfaces of constant ̃ηand constant ̃χ
can then be drawn in theη–χplane, whereη,χare related to ̃η,χ ̃as in (2.41) with
the substitutionst→η ̃andr→χ. ̃ In open and flat universes, the lower boundary
(η ̃= 0 )corresponds to a physical singularity.